Lightning surges in hybrid cable-overhead lines: Part II—voltage estimation for strikes to shield wire

Abstract

This paper (Part II) continues the study from Part I by presenting formulas able to do a fast estimation of the voltage at an underground cable sheath for lightning surges striking an earth wire, with and without back-flashover. Similar to Part I, these formulas do not require an electromagnetic transients (EMT) software and can be implemented as scrip for a fast screening of potential issues, requiring solely the geometric data of the cable and the overhead line. The potential impact and limitations of the simplifications are evaluated.

Appendix

All simulations were done in PSCAD/EMTDC. The cables and OHLs were modeled using geometric frequency-dependent models, with the data from Tables 1 and 2, respectively. Figure 5 shows the position of the phase conductors and earth wires for the OHL, as well as the thickness and position of the core, insulation, sheath and outer insulation for the cable. Figure 6 shows the schematic of the simulated cases, where C₂ is without back-flashover and C_2.2 is with back-flashover.

Table 1 Cable data

Table 2 OHL data

Fig. 5

Geometric data on OHL (left) and cable (right)

Fig. 6

Schematic for the simulations: C₂ corresponds to striking on earth wire and no back-flashover; C_2.2 to striking on earth wire and back-flashover. The three phases of the OHL are represented as one line, with the back-flashover being to only one phase. The current variables are the ones used for the case with back-flashover (10)

Equation (17) shows the calculation of the sheath voltage at the transition point by combining (10) with (15).

$$\begin{aligned} \frac{{V_{{{\text{EW}}\_I}} }}{{Z_{{{\text{EW}}}} }} - \frac{{V_{S} - V_{{{\text{EW}}\_I}} }}{{Z_{{{\text{EW}}}} }} = 2\frac{{V_{S} }}{{Z_{S} }}\left( {\frac{{Z_{S} }}{{Z_{G} + Z_{S} }}} \right)\\ \quad + \frac{{V_{S} }}{{Z_{G} }} - \frac{{2V_{{{\text{EW}}\_I}} }}{{Z_{C} + Z_{P} }}\left( {\frac{{Z_{S} }}{{Z_{G} + Z_{S} }}} \right) \hfill \\ \Leftrightarrow \frac{{2V_{{{\text{EW}}\_I}} }}{{Z_{{{\text{EW}}}} }} + \frac{{2V_{{{\text{EW}}\_I}} }}{{Z_{C} + Z_{P} }}\left( {\frac{{Z_{S} }}{{Z_{G} + Z_{S} }}} \right)\\ = 2\frac{{V_{S} }}{{Z_{S} }}\left( {\frac{{Z_{S} }}{{Z_{G} + Z_{S} }}} \right) + \frac{{V_{S} }}{{Z_{G} }} + \frac{{V_{S} }}{{Z_{{{\text{EW}}}} }} \hfill \\ \Leftrightarrow \frac{{2V_{{{\text{EW}}\_I}} Z_{S} Z_{G} }}{{Z_{{{\text{EW}}}} Z_{S} Z_{G} }} + \frac{{2V_{{{\text{EW}}\_I}} }}{{Z_{C} + Z_{P} }}\left( {\frac{{Z_{S} }}{{Z_{G} + Z_{S} }}} \right)\\ = \frac{{V_{S} }}{{Z_{{{\text{EW}}}} Z_{S} Z_{G} }}\left( {2Z_{{{\text{EW}}}} Z_{G} \left( {\frac{{Z_{S} }}{{Z_{G} + Z_{S} }}} \right) + Z_{{{\text{EW}}}} Z_{S} + Z_{S} Z_{G} } \right) \hfill \\ \Leftrightarrow \frac{{2V_{{{\text{EW}}\_I}} Z_{S} Z_{G} }}{{Z_{{{\text{EW}}}} Z_{S} Z_{G} }} + \frac{{2V_{{{\text{EW}}\_I}} }}{{Z_{C} + Z_{P} }}\left( {\frac{{Z_{S} }}{{Z_{G} + Z_{S} }}} \right)\\ = \frac{{V_{S} }}{{Z_{{{\text{EW}}}} Z_{S} Z_{G} \left( {Z_{G}+ Z_{S} } \right)}}\\ \times( 2Z_{{{\text{EW}}}} Z_{G} Z_{S} + Z_{{{\text{EW}}}} Z_{S} \left( {Z_{G} + Z_{S} } \right)+ Z_{S} Z_{G} \left( {Z_{G} + Z_{S} } \right) ) \hfill \\ \Leftrightarrow \frac{{2V_{{{\text{EW}}\_I}} Z_{G} }}{{Z_{{{\text{EW}}}} Z_{G} }} + \frac{{2V_{{{\text{EW}}\_I}} }}{{Z_{C} + Z_{P} }}\left( {\frac{{Z_{S} }}{{Z_{G} + Z_{S} }}} \right)\\ = \frac{{V_{S} }}{{Z_{{{\text{EW}}}} Z_{G} \left( {Z_{G} + Z_{S} } \right)}}( 2Z_{{{\text{EW}}}} Z_{G} + Z_{{{\text{EW}}}} \left( {Z_{G} + Z_{S} } \right)\\ \quad + Z_{G} \left( {Z_{G} + Z_{S} } \right) ) \hfill \\ \Leftrightarrow 2V_{{{\text{EW}}\_I}} Z_{G} \left( {Z_{C} + Z_{P} } \right)\left( {Z_{G} + Z_{S} } \right) + 2V_{{{\text{EW}}\_I}} Z_{{{\text{EW}}}} Z_{S} Z_{G}\\ = V_{S} \left( {2Z_{{{\text{EW}}}} Z_{G} + Z_{{{\text{EW}}}} \left( {Z_{G} + Z_{S} } \right) + Z_{G} \left( {Z_{G} + Z_{S} } \right)} \right)\left( {Z_{C} + Z_{P} } \right) \hfill \\ \Leftrightarrow V_{S} = V_{{{\text{EW}}\_I}} \frac{{2Z_{G} \left( {Z_{C} + Z_{P} } \right)\left( {Z_{G} + Z_{S} } \right) + 2Z_{{{\text{EW}}}} Z_{S} Z_{G} }}{{\left( {2Z_{{{\text{EW}}}} Z_{G} + Z_{{{\text{EW}}}} \left( {Z_{G} + Z_{S} } \right) + Z_{G} \left( {Z_{G} + Z_{S} } \right)} \right)\left( {Z_{C} + Z_{P} } \right)}} \hfill \\ \end{aligned} $$

(17)